We are asked to verify the identity:

$\tan \left( x + \frac{\pi}{2} \right) = -\cot(x)$

Step-by-step verification:

1. Use the tangent addition formula: $\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}$ For $a = x$ and $b = \frac{\pi}{2}$, we have: $\tan\left(x + \frac{\pi}{2}\right) = \frac{\tan(x) + \tan\left(\frac{\pi}{2}\right)}{1 - \tan(x)\tan\left(\frac{\pi}{2}\right)}$

2. Evaluate $\tan\left(\frac{\pi}{2}\right)$: $\tan\left(\frac{\pi}{2}\right) \text{ is undefined because it goes to infinity}.$ However, when substituting it into the formula, we treat it carefully as follows:

   $\tan\left(\frac{\pi}{2}\right) = \infty$

   Substituting $\infty$ in, the expression simplifies to:

   $\frac{\tan(x) + \infty}{1 - \tan(x)\cdot\infty}$

3. Simplifying the expression: As $\infty$ is involved, the whole fraction simplifies. The numerator becomes dominated by $\infty$, and the denominator becomes very negative as $\tan(x) \cdot \infty$ approaches infinity negatively: $\tan\left(x + \frac{\pi}{2}\right) = -\frac{1}{\tan(x)} = -\cot(x)$

Thus, the identity is verified:

$\tan \left( x + \frac{\pi}{2} \right) = -\cot(x)$

Conclusion:

The given identity is correct.

Would you like more details or have any further questions?

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Here are 5 related questions to expand on the concept:

1. What is the sine addition formula and how is it used?
2. Can you prove the identity $\cot(x + \frac{\pi}{2}) = -\tan(x)$?
3. What is the relationship between tangent and cotangent functions?
4. How does the behavior of trigonometric functions change with phase shifts like $\frac{\pi}{2}$?
5. What are the key differences between trigonometric and inverse trigonometric identities?

Tip: Memorizing the basic trigonometric identities, such as tangent and cotangent relationships, can help simplify more complex problems!